---
tag: DC60FYCSPM
kind: concept
name: Second-order arithmetic
msc: [03F35, 03B30]
sources:
  - url: https://en.wikipedia.org/wiki/Second-order_arithmetic
same_as:
  wikidata: Q7442973
  wikipedia: Second-order arithmetic
---

## Summary

Second-order arithmetic, often written $\mathsf{Z}_2$, is a theory with two kinds of variables.
One kind ranges over natural numbers, and the other over sets of natural numbers.
Its axioms are basic facts about arithmetic, induction for sets, and comprehension, which says that every definable collection of numbers is a set.
A large part of ordinary mathematics, including real analysis, can be developed in it by coding real numbers and other objects as sets of numbers.
Its subsystems are the setting of [[reverse mathematics|7SQS69T17R]].

## Explanations

### L1 human

First-order arithmetic talks only about numbers, such as $3$ or $17$.
Second-order arithmetic can also talk about sets of numbers, such as the set of all primes.
A real number can be coded as a set of numbers, for example by a sequence of rational approximations, so the theory can talk about real numbers too.
This makes it strong enough for most of calculus and analysis, while it stays a theory about numbers at heart.